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The field of the invention is nuclear magnetic resonance imaging (xe2x80x9cMRIxe2x80x9d) methods and systems. More particularly, the invention relates to systems and methods for increasing the signal to noise ratio in MRI images where image data has to be unwrapped during data processing.
When a substance such as human tissue is subjected to a uniform magnetic field (polarizing field B0), the individual magnetic moments of the spins in the tissue attempt to align with this polarizing field, but precess about it in random order at their characteristic Larmor frequency. If the substance, or tissue, is subjected to a magnetic field (excitation field B1) which is in the x-y plane and which is near the Larmor frequency, the net aligned moment, Mz, may be rotated, or xe2x80x9ctippedxe2x80x9d, into the x-y plane to produce a net transverse magnetic moment Mt. A signal is emitted by the excited spins after the excitation signal B1 is terminated, this signal may be received by receiver coils and processed to form an image.
When utilizing these signals to produce images, magnetic field gradients (Gx Gy and Gz) are employed to select locations within the tissue for excitation. Typically, the region to be imaged is scanned by a sequence of measurement cycles in which gradients Gx Gy and Gz vary according to the particular localization method being used. Herein it will be assumed that gradient Gy is used to adjust the phase of signals along a phase encoding axis Y. The resulting set of received nuclear magnetic resonance (NMR) signals are digitized and stored in a k-space raster format. An exemplary k-space raster 10 is illustrated in FIG. 2 and includes a plurality of rows 12 of data. After all of the k-space data has been acquired, the data is typically subjected to a two-dimensional Fourier transform and the resulting data is then used to reconstruct an image using one of several different reconstruction techniques. An exemplary resulting image 14 is also illustrated in FIG. 2.
Thus, the intensity of each pixel in an MR image is generally a function of two factors. First, pixel signal intensity is a function of the spin density m at a point in an object slice being imaged that corresponds to the particular pixel in the image. Second, pixel signal intensity is also a function of the operating characteristics of the receiving coil that receives the NMR signal and converts the signal to an analog signal and then to a digital signal for storage in k-space. Specifically, the coil operating characteristic that affects the end pixel intensity the most is referred to as coil sensitivity s. Thus, intensity i for a pixel y can be expressed by the following Equation:
i(y)=s(y)m(Y)xe2x80x83xe2x80x83Equation 1
There are several different factors that can be used to judge the value of any imaging system but two of the most important factors are the quality of the resulting images and the speed with which imaging data can be acquired. Higher quality images increase diagnostic value. Acquisition speed increases system throughput (i.e., the number of imaging sessions that can be performed in a given period) and can also increase image quality as patient movement is reduced when the acquisition period is chortened (i.e., patient movement is less likely during a shorter period than during a longer period. With MRI systems, throughput is extremely important as MRI systems are relatively expensive and the expense is in part justified by the amount of use a system receives.
One way to increase system throughput is to reduce the amount of data collected during an imaging session. For example, one way to reduce the amount of collected data is to increase the space between phase encoding lines in k-space. Referring to FIG. 3, an exemplary k-space raster 20 having half as many k-space lines as the raster 10 of FIG. 2 is illustrated. The time required to collect the data in raster 20 would be approximately half the time required to collect the data in raster 10.
By reducing the number of phase encoding lines employed during data acquisition, the field of view (FOV) along the phase encoding axis Y of the resulting image is also reduced. Referring again to FIG. 3, the FOV for the image 15 in FIG. 3 is shown as being approximately xc2xd the FOV in the image of FIG. 2 along the phase encoding axis Y. Where the object being imaged fits within the reduced FOV, the reduced FOV does not affect the resulting image. Because reducing the k-space phase encoded lines reduces the FOV, the factor by which the k space phase encoding lines are reduced is referred to as the reduction factor R.
Unfortunately, where the object being imaged extends outside the reduced FOV, the image sections that correspond to the out-of-FOV object sections xe2x80x9cwrap aroundxe2x80x9d on the image and are overlaid on other image sections. Thus, in FIG. 3, out-of-FOV image sections 22 and 24 wrap around and are overlaid on in-FOV image section 25 thereby generating wrapped sections 28 and 26, respectively. Each wrapped section (e.g., 22) includes pixel intensities that are the sum of two intensities corresponding to two different pixels in a non-wrapped image (i.e., in an image like that of FIG. 2). The two intensities that combine to produce each wrapped pixel intensity include one intensity corresponding to an in-FOV pixel and one intensity corresponding to an out-of-FOV pixel. For example, referring to FIGS. 2 and 3, the FOV of FIG. 3 is also illustrated in FIG. 2 by the space between lines 36 and 38 and thus when the FOV is reduced as in FIG. 3, pixel 30 is an in-FOV pixel and pixel 40 is an out-of-FOV pixel. Thus the pixel intensity of pixel 40 wraps as indicated by arrow 41 and is laid over pixel 30 intensity. Together the intensities of pixels 30 and 40 add to generate the intensity of pixel 42 upon wrapping.
Where the reduction factor R is greater than 2 additional image wrapping can occur thereby causing wrapped image pixels to include intensity corresponding to more than two (e.g., 3, 4, etc.) unwrapped pixels. This additional wrapping further reduces the diagnostic value of the resulting image.
The industry has devised ways to effectively xe2x80x9cunwrapxe2x80x9d wrapped images like the exemplary image in FIG. 3. It has been recognized that by providing several NMR signal receiving coils where the sensitivities of each coil are known, a permutation of Equation 1 above can be used to separate the intensity of a wrapped pixel into the in-FOV intensity and the out-of-FOV intensity. To this end, along a phase encoding axis the intensity of a wrapped pixel y corresponding to first and second receiver coils can be expressed as:
i1(y)=s1(y)m(y) +s1(y+D)m(y+D)xe2x80x83xe2x80x83Equation 2
i2(y)=s2(y)m(y) +s2(y+D)m(y+D)xe2x80x83xe2x80x83Equation 3
respectively, where D is the phase encoding FOV (see FIG. 3).
Referring still to Equations 2 and 3, assuming that the sensitivities s1 and s2 for each of the first and second coils are known, after intensity data has been acquired for each of the first and second coils, only m(y) and m(y+D) are unknown. Thus, Equations 2 and 3 can be solved for each of the unknowns to determine the spin densities m(y) and m(y+D) at pixels y and y+D, respectively. The spin densities at each pixel can then be converted to intensities to xe2x80x9cunwrapxe2x80x9d the image.
By increasing the reduction factor R, the amount of data acquired is reduced and therefore throughput is accelerated. The number of unknowns, however, that can be resolved in any system is equal to the number of separate receiver coils in the system. Thus, in any given system the maximum reduction factor R is equal to the number of receiver coils N. For example, in the exemplary system described above that includes four receiver coils the maximum reduction factor R is 4.
In general terms, the intensity of a particular wrapped pixel y with a reduction factor R can be expressed by the equation:                                           i            j                    ⁡                      (            y            )                          =                              ∑                          k              =              0                                      R              -              1                                ⁢                                                    s                j                            ⁡                              (                                  y                  +                                      k                    ⁢                                          xe2x80x83                                        ⁢                    d                                                  )                                      ⁢                          xe2x80x83                        ⁢            m            ⁢                          xe2x80x83                        ⁢                          (                              y                +                                  k                  ⁢                                      xe2x80x83                                    ⁢                  D                                            )                                                          Equation        ⁢                  xe2x80x83                ⁢        4            
where j refers to coil number and sj(y) refers to the sensitivity of coil j.
In a system including N coils, intensity, spin density and sensitivity matrices I, M and S can be defined as having dimensions Nxc3x971, Rxc3x971 And Nxc3x97R, respectively, and Equation 4 can be rewritten in matrix form as:
I=SMxe2x80x83xe2x80x83Equation 5
Again, assuming S is known, S can be inverted and Equation 5 can be rewritten and solved for an estimated spin density M as:
{circumflex over (M)}=Sxe2x88x921I xe2x80x83xe2x80x83Equation 6
A solution of Equation 6 that can be relied upon should minimize the spin density estimate error. Thus, a typical solution to Equation 6 minimizes |S{circumflex over (M)}xe2x88x92I|2 resulting in the following equation:
xe2x80x83{circumflex over (M)}=[(S+S)xe2x88x921S+]Ixe2x80x83xe2x80x83Equation 7
where + denotes the Hermitian conjugate.
Obviously the value of the solution to Equation 7 is only as good as the preciseness with which the sensitivities of the coils can be determined. Unfortunately, while the industry has devised several ways to determine coil sensitivities, there are many factors that affect sensitivities such that noise often occurs upon inversion of the sensitivity matrix S that propagates and is exacerbated in resulting images. Thus, while solving Equation 7 in theory provides a way to unwrap image data, in practice the resulting image has a relatively low signal to noise ratio (SNR).
The present invention is a method for reducing the noise in an unwrapped image by modifying the coil sensitivity matrix and using the modified matrix instead of the original matrix to determine the spin densities of image pixels by solving an equation similar to Equation 7 above.
More specifically, one way to increase the SNR is to minimize the function A({circumflex over (M)})+xcexB({circumflex over (M)}) where A({circumflex over (M)})=|S{circumflex over (M)}xe2x88x92I|2, xcex is an adjustable parameter and B({circumflex over (M)}) is chosen to reduce system sensitivity to noise. For some square matrix H, B({circumflex over (M)}) may have the form B({circumflex over (M)})={circumflex over (M)}+H{circumflex over (M)} where, again the + indicates the Hermitian conjugate. With B({circumflex over (M)}) expressed in the above form, Equation 7 can be rewritten as:
{circumflex over (M)}=[(S+S+xcexH)xe2x88x921S+]Ixe2x80x83xe2x80x83Equation 8
Because high noise often results in large pixel values a simple solution to Equation 8 is to minimize the magnitude of {circumflex over (M)} so that B({circumflex over (M)})={circumflex over (M)}+{circumflex over (M)} giving H=U where U is the unit matrix. Thus, Equation 8 can be rewritten as:
{circumflex over (M)}=[(S+S+xcexU)xe2x88x921S+]Ixe2x80x83xe2x80x83Equation 9 
Upon solving Equation 9 the resulting spin density estimates {circumflex over (M)} can be used to generate an unwrapped image with relatively high SNR.
In addition to several methods the invention also includes an apparatus for performing each of the methods.
These and other aspects of the invention will become apparent from the following description. In the description, reference is made to the accompanying drawings which form a part hereof, and in which there is shown a preferred embodiment of the invention. Such embodiment does not necessarily represent the full scope of the invention and reference is made therefore, to the claims herein for interpreting the scope of the invention.